First, consider the ramifications to the user by auto-printing, especially in a corporate environment. The expected behavior is to load the PDF then the user can save it, or send it to a printer of their choice. Most people save PDF’s today versus printing.
You may want to look at how to print an external page without opening it at
I have not used this technique with a PDF, only with content generated by a web app.
Hope this helps you make a decision or leads you to a solution that works for you.